Systematic sampling
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General descriptions of systematic sampling
Systematic sampling is a name of a wide class of sampling strategies in which selection of individual elements is following a systematic pattern. Examples are square grids of sample points laid out over an area of interest; or the selection of every 10th tree in an alley; or parallel transects.
Systematic sampling and its applications to forest inventory are best illustrated with square grids of sample points. We may imagine a transparency sheet on which this grid is printed; and this transparency is placed randomly over the map, where randomly means: randomly selected starting point and random orientation. From a sample selection point of view, it is important to state that we have only one independent selection of a sample point; after having selected the first point, all others are fixed. We defined earlier that sample size is the number of independently selected elements; an immediate conclusion is that systematic sampling is obviously a sample of size \(n = 1\). The “plot” that is being laid out then is a large cluster plot consisting of numerous sub-plots – that is, all the sample points on the systematic sample are strictly spoken sub-plots of one single cluster that is spread out over the entire area.
A major question is then whether we can make an unbiased estimation of mean and variance from a random sample of size \(n = 1\). For the estimation of the mean, there is no problem at all: the estimator
\(\bar y=\frac{\sum_{i=1}^n y_i}{}n\,\)
can be calculated and yields the estimation of the mean.
However, when we wish to estimate the variance with the estimator
\(s^2=\frac{\sum_{i=1}^n\left(y_i-\bar y\right)^2}{n-1}\,\),
we see that this is not possible as the denominator is not defined. This is also directly understandable by common sense: one single observation does not contain any information about the variability that is present in the population.
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